Acoustic cloaking of spherical objects using thin elastic coatings

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Copyright by

Matthew David Guild 2012

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The Dissertation Committee for Matthew David Guild

certifies that this is the approved version of the following dissertation:

Acoustic Cloaking of Spherical Objects Using Thin Elastic Coatings

Committee:

Andrea Al`u, Supervisor

Michael R. Haberman, Supervisor

Mark F. Hamilton

Preston S. Wilson

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by

Matthew David Guild, B.S.; M.S.E.

DISSERTATION

Presented to the Faculty of the Graduate School of The University of Texas at Austin

in Partial Fulfillment of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN May 2012

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Publication No.

Matthew David Guild, Ph.D. The University of Texas at Austin, 2012

Supervisors: Andrea Al`u

Michael R. Haberman

In this thesis, a detailed description of acoustic cloaking is put forth using a coating consisting of discrete layers, enabling the cancellation of the scattered field around the object. This particular approach has previously only been applied to electromagnetic waves, for which it was observed that cloaking could be achieved using isotropic materials over a finite bandwidth. The analysis begins with a presen-tation of the theoretical formulation, which is developed using classical scattering theory for the scattered acoustic field of an isotropic sphere coated with multiple layers. Unlike previous works on acoustic scattering from spherical bodies, the cri-teria for acoustic cloaking is that the scattered field in the surrounding medium be equal to zero, and seeking a solution for the layer properties which achieve this condition.

To effectively investigate this situation, approximate solutions are obtained by assuming either quasi-static limits or thin shells, which provide valuable insight into the fundamental nature of the scattering cancellation. In addition, using these approximate solutions as a guide, exact numerical solutions can be obtained, en-abling the full dynamics of the parameter space to be evaluated. Based on this

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analysis, two distinct types of acoustic cloaking were found: a plasmonic cloak and an anti-resonance cloak.

The plasmonic cloak is a non-resonant type of cloak, named plasmonic be-cause of its analogous behavior to the non-resonant cloak observed in electromag-netic waves which utilizes plasmonic materials to achieve the necessary properties. Due to the non-resonant behavior, this type of cloak offers the possibility of a much broader range of cloaking. To expand this design beyond wavelengths on the order of the uncloaked scatterer, multilayered cloak designs are investigated.

The anti-resonance cloak, as the name suggests, uses the anti-resonances of the modes within the cloaking layer to supplement the non-resonant plasmonic cloaking of the scattered field. Although somewhat more limited in bandwidth due to the presence of anti-resonances (and the accompanying resonances), this type of cloak enables a larger reduction in the scattering strength, compared with using a single elastic layer utilizing only non-resonant cloaking. A thorough investigation of the design space for a single isotropic elastic cloaking layer is performed, and the necessary elastic properties are discussed.

The work in this thesis describes the investigation of the theoretical for-mulation for acoustic cloaking, expanding upon the use of scattering cancellation previously developed for the cloaking of electromagnetic waves. This work includes a detailed look at the different physical phenomena, including both resonant and non-resonant mechanisms, that can be used to achieve the necessary scattering can-cellation and which can be applied to a wide range of scattering configurations for which cloaking would be desirable. In addition to laying out a broad theoretical foundation, the use of limiting cases and practical examples demonstrates the effec-tiveness and feasibility of such an approach to the acoustic cloaking of a spherical object.

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List of Tables

4.1 Properties of elastic spheres to be cloaked for the three examples considered in this section. . . 80 4.2 Properties of single fluid layer plasmonic cloak of an elastic core for

ka = 0.5, 0.75, and 1.00. . . 81 4.3 Properties of single fluid layer plasmonic cloak of a fluid core for

ka = 0.5, 0.75, and 1.00. . . 82 5.1 Material properties of a single elastic cloaking layer designed at kd,0a =

1.0 with b

a= 1.30 for a rigid, immovable sphere. . . 104 5.2 Material properties of a single elastic cloaking layer designed at kd,0a =

1.0 with b

a= 1.30 for a rigid, immovable sphere, based on the first anti-resonance of the n = 1 mode. . . 108 5.3 Material properties of spheres to be cloaked in Sections 5.3 and 5.4. 115 5.4 Properties of a single elastic cloaking layer with b

a = 1.30 for the different core materials examined in Sections 5.3 and 5.4. . . 116 6.1 Cloaking layer properties for a steel sphere in water, coated by an

acoustic plasmonic cloak consisting of two layers with shell thicknesses δ1= δ2= 0.04 and a design frequency of kd,0a = 2.0. Solutions are given based on analytic thin-shell expressions, exact solutions for the case of two fluid layers, and exact solutions for the case of a fluid inner layer and isotropic elastic outer layer with νc1= 0.3. . . 152 6.2 Cloaking layer properties for a four layer acoustic plasmonic cloak

for a steel sphere in water at a design frequency of kd,0a = 2.0. The shell thicknesses δ of the four layers, in order of the outermost to innermost layer, are 0.0135, 0.0642, 0.0037 and 0.0046, respectively. Exact solutions obtained numerically for the case of four fluid layers, and the case of alternating fluid and isotropic elastic layers with νc1= 0.3. . . 161

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List of Figures

2.1 Illustration of the coordinate transformation between an undeformed space (left) and a symmetric deformed space (right). Any object placed within the interior region of radius a will be cloaked. . . 8 2.2 (a) An incident plane wave impinging upon a cylindrical body covered

with an inertial cloak with density tensor ρ and bulk modulus κ. (b) Numerical simulation from Cummer et al. of the total pressure field for a perfect inertial cloak [14]. The amplitude is normalized by the magnitude of the incident wave, which is a time-harmonic plane wave traveling from left to right. . . 10 2.3 (a) Schematic view of a cloaked cylinder proposed by Torrent and

S´anchez-Dehesa [15]. The cloak consists of alternating fluid sublayers of equal thickness. (b) Parametric plot showing the range of mate-rial properties needed to achieve cloaking using a configuration like that shown in part (a). The solid lines show the two fluid sublayer properties, and the dashed lines show the design space using a sonic crystal arrangement. . . 15 2.4 Numerical simulations of the total pressure field for a cloaked rigid

cylinder, using an inertial cloak consisting of alternating fluid sub-layers containing sonic crystals by Torrent and S´anchez-Dehesa [15], with (a) 50 layers, and (b) 200 layers. . . 16 2.5 Experimental design by Zhang, Xia and Fang [30] for a cloaked steel

cylinder in water, using an inertial cloak with 16 discrete layers of a radially symmetric lattice of ports and cavities, for (a) the entire cloak, and (b) a close up to show the arrangement of ports and cav-ities. The measured pressure field passing through the shadow zone of the rigid cylinder are shown in (c) for the cloaked (blue) and un-cloaked (red) configurations, relative to a freefield measurement (green). 17 2.6 (a) Example of an acoustic metafluid by Norris [21], consisting of

lubricated elliptical beads arranged in a hexagonal lattice. (b) A pentamode cloak using a proposed acoustic metafluid called metal water [37]. . . 21

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2.7 (a) Numerical simulation by Nicorovici et al. [39] of the electric po-tential field demonstrating anomalous resonance cloaking. The inner and outer edge of the cloak are denoted by solid lines, and the re-gion of cloaking is denoted with a dashed line. (b) Layout of a 1D mass-spring system by Fang et al. [41] experimentally shown to ex-hibit negative effective mass. (c) Proposed resonant inclusion for an acoustic anomalous resonance cloak by Zhou et al. [42]. In this illus-tration, red denotes upward displacement and blue denotes downward displacement. . . 23 2.8 (a) Demonstration of EM scattering cancellation using a plasmonic

cloak [44], originally developed by Al`u and Engheta [45]. (b) Para-metric plot for the cloaking layer properties of a single layer cloak utilizing EM scattering cancellation for a magnetodielectric sphere [44]. The color scale represents the scattering strength relative to the uncloaked scatterer. . . 26 3.1 A time-harmonic incident plane wave in a fluid medium impinging on

an isotropic elastic core of radius a coated with multiple shells with outer radius b. The surrounding medium has density ρ0 and bulk modulus κ0, and the elastic core has density ρ, bulk modulus κ and shear modulus µ. . . 36 3.2 Schematic of the minimization algorithm used to find the properties

and geometry of the cloak using the scattering cancellation approach. 47 3.3 Parametric plot of the optimized cloaking layer density (top panel),

cloaking layer bulk modulus (middle panel) and scattering gain (bot-tom panel) as a function of the density and bulk modulus of a pen-etrable fluid scatterer relative to the external fluid. The results are given for a cloaking layer with a thickness ratio of b/a = 1.10 at ka = 0.5. . . 49 4.1 A time-harmonic incident plane wave in a fluid medium impinging

on an isotropic elastic core of radius a coated with a single fluid shell with outer radius b. The surrounding medium has density ρ0 and

bulk modulus κ0, the fluid shell has density ρcand bulk modulus κc, and the elastic core has density ρ, bulk modulus κ and shear modulus µ. . . 51 4.2 Magnitude of the spherical Bessel functions (top) which make up

the numerator and denominator of the scatter coefficients for a rigid, immovable sphere (bottom). The magnitude of the scattering coeffi-cients are given in dB. . . 67

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4.3 Variation of the cloaking layer parameters as a function of the ratio of the outer radius of the cloaking layer b to the outer radius of the scatterer a. The cloaking layer properties, given by ρc/ρ0(top panel) and κc/κ0 (middle panel), represent the values which cancel the first two modes at kd,0a = 0.5 using a single fluid plasmonic cloak coating a rigid, immovable sphere. The bottom panel gives the scattering gain in dB, relative to the uncloaked scatterer. . . 70 4.4 Scattering coefficients (in dB) for an uncloaked (top panel) and cloaked

(middle panel) rigid, immovable sphere. The cloak consists of a single fluid layer with _ab= 1.10 which cancels the first two scattering modes at kd,0a = 0.5. The scattering gain in dB, relative to the uncloaked scatterer, is given in the bottom panel . . . 72 4.5 Scattering coefficients (in dB) for an uncloaked (top panel) and cloaked

(middle panel) rigid, immovable sphere. The cloak consists of a single fluid layer with _ab= 1.10 which cancels the first two scattering modes at kd,0a = 0.5. Results for loss factors of γ = 0 (lossless), γ = 0.001, γ = 0.01 and γ = 0.1 are shown. The scattering gain in dB, relative to the uncloaked scatterer, is given in the bottom panel. . . 73 4.6 Parametric plot of the optimized cloaking layer density (top panel),

cloaking layer bulk modulus (middle panel) and scattering gain (bot-tom panel) as a function of the density and bulk modulus of a pen-etrable fluid scatterer relative to the external fluid. The results are given for a cloaking layer with b/a = 1.10, which cancels the first two modes at kd,0a = 0.5. . . 76 4.7 Slices of the parametric plots of Figure 4.6 for constant scatterer bulk

modulus of κ/κ0 = 1 (left column) and scatterer density of ρ/ρ0= 1 (right column). The rows depict resulting changes in the cloaking layer density (top row), cloaking layer bulk modulus (middle row) and scattering gain (bottom row). . . 78 4.8 Scattering gain as a function of ka for a cloaked sphere of stainless

steel (top), aluminum (middle) and glass (bottom). The scattering gain is given in dB relative to the scattering strength of the uncloaked sphere. Three plasmonic cloaking layers are presented for each case, optimized for ka = 0.5 (red), ka = 0.75 (blue) and ka = 1.0 (green). The cloaking layer material properties for each case are listed in Ta-bles 4.2 and 4.3. The cloaking layer thickness ratio in all cases was b/a = 1.05. . . 84 4.9 Comparison of the scattering coefficients for an elastic glass sphere

(right column), and a fluid glass sphere (left column). For each case, the first 5 scattering coefficients are given for the uncloaked sphere (top row) and cloaked sphere (bottom row). The cloaking layer prop-erties are given by Table 4.2 for the elastic glass sphere, and Table 4.3 for the fluid glass sphere. . . 85

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4.10 Real part of total pressure field for an isotropic sphere of stainless steel (left column), aluminum (middle column) and glass (right col-umn). For each case, the uncloaked sphere is presented in the top row, and with a plasmonic cloak consisting of a single fluid layer with a thickness ratio of b/a = 1.05 presented in the bottom row. The cloaking layer properties for each case are listed in Table 4.2. The color scale for the pressure is normalized to the amplitude of the inci-dent wave, which is a time-harmonic plane wave traveling from left to right with a frequency of kd,0a = 1. The length scale r is normalized by the uncloaked sphere radius a. . . 86 4.11 Parametric plot of elastic effects on the cloaking layer parameter κc

in the quasi-static limit, as a function of the cloaking layer Poisson’s ratio νc and the core material bulk modulus κ, obtained from Equa-tion (4.88). The different colors represent the (dimensionless) values of κcon a logarithmic scale. . . 92 4.12 Scattering coefficients (in dB) for an uncloaked (top) and cloaked

(bottom) pressure-release sphere. The cloak consists of a single elastic layer with νc= 0.3 and _ab= 1.10, which cancels the first two scattering modes at kd,0a = 0.5. . . 95 5.1 A time-harmonic incident plane wave in a fluid medium impinging

on an isotropic elastic core of radius a coated with a single isotropic elastic shell with outer radius b. The surrounding medium has density ρ0 and bulk modulus κ0, and the elastic shell has density ρc, bulk modulus κc, and shear modulus µc. The elastic core has density ρ, bulk modulus κ and shear modulus µ. . . 98 5.2 Parametric study of the solutions satisfying |Un| = 0 for a rigid,

im-movable sphere coated in a single elastic cloaking layer with νc= 0.49 and b

a = 1.30 at kd,0a = 1.0. In (a), |U0| = 0 is solved for κc as a function of ρ_c. In (b), the magnitude of |U1| and |V1| are plotted as a function of ρ_c. . . 103 5.3 Magnitude of the scattering coefficients (in dB) for a rigid, immovable

sphere: (a) uncloaked, (b) cloaked using the first anti-resonance, (c) cloaked using the second anti-resonance, and (d) cloaked using the third anti-resonance. All the cloaks consist of a single elastic layer with νc= 0.49 and

b

a= 1.30, designed to cancel the first two modes at kd,0a = 1.0, with the material properties listed in Table 5.1. . . 105

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5.4 Parametric study of the solutions satisfying |Un| = 0 for a rigid, im-movable sphere coated in a single elastic cloaking layer with b

a= 1.30 at kd,0a = 1.0. In (a), |U0| = 0 is solved for κc as a function of ρc. In (b), the magnitude of |U1| is plotted as a function of ρc. Results are shown for νc= 0.495 (red), νc= 0.490 (blue), νc= 0.450 (green), and νc= 0.300 (magenta). . . 107 5.5 Magnitude of the scattering coefficients (in dB) for a rigid, immovable

sphere cloaked using a single elastic layer with b

a = 1.30 and: (a) νc= 0.3, (b) νc= 0.45, (c) νc= 0.49. The cloaks are designed to cancel the first 2 modes at kd,0a = 1.0, with the material properties listed in Table 5.2. The scattering gain in dB, relative to the uncloaked scatterer, is given for (d) νc= 0.3, (e) νc= 0.45, (f) νc= 0.49. . . 110 5.6 Parametric study of the solutions satisfying |Un| = 0 for a rigid,

im-movable sphere coated in a single fluid cloaking layer with b

a= 1.30 at kd,0a = 1.0. In (a), |U0| = 0 is solved for κc as a function of ρc. In (b), the magnitude of |U1| and |V1| are plotted as a function of ρc. . 111 5.7 Magnitude of the scattering coefficients (in dB) for a rigid, immovable

sphere: (a) uncloaked, and (b) cloaked using a single fluid layer with b

a= 1.30, designed to cancel the first two modes at kd,0a = 1.0. . . . 113 5.8 Parametric study of the solutions satisfying |Un| = 0 for a stiff,

neu-trally buoyant sphere coated in a single elastic cloaking layer with νc= 0.4884 and

b

a= 1.30 at kd,0a = 1.0. In (a), |U0| = 0 is solved for κc as a function of ρ_c. In (b), the magnitude of |U1| and |V1| are plotted as a function of ρ_c. In (c), the magnitude of |U2| and |V2| are plotted as a function of ρ_c. . . 114 5.9 Magnitude of the scattering coefficients (in dB) for a stiff, neutrally

buoyant sphere: (a) uncloaked, and (b) cloaked using a single elas-tic layer with νc= 0.4884 and

b

a= 1.30, designed to cancel the first two modes at kd,0a = 1.0, with the material properties listed in Ta-ble 5.4. The scattering gain in dB, relative to the uncloaked scatterer, is given in (c) for the exact theoretical solution and using finite ele-ments (COMSOL). . . 117

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5.10 Real part of total pressure field for a stiff, neutrally buoyant sphere: (a) uncloaked, and (b) cloaked using a single elastic layer with νc= 0.4884 and b

a= 1.30, designed to cancel the first two modes at kd,0a = 1.0, with the material properties listed in Table 5.4. The color scale for the pressure is normalized to the amplitude of the incident wave, which is a time-harmonic plane wave traveling from bottom to top with a frequency of kd,0a = 1.0. . . 118 5.11 Parametric study of the solutions satisfying |Un| = 0 for a hollow

(pressure-release) sphere coated in a single elastic cloaking layer with νc= 0.3 and

b

a= 1.30 at kd,0a = 1.75. In (a), |U0| = 0 is solved for κc as a function of ρ_c. In (b), the magnitude of |U1| and |V1| are plotted as a function of ρ_c. In (c), the magnitude of |U2| and |V2| are plotted as a function of ρ_c. . . 119 5.12 Magnitude of the scattering coefficients (in dB) for a hollow

(pressure-release) sphere: (a) uncloaked, and (b) cloaked using a single elas-tic layer with νc = 0.3 and

b

a = 1.30, designed to cancel the first 4 modes at kd,0a = 1.75, with the material properties listed in Ta-ble 5.4. The scattering gain (in dB) is given in (c), referenced to the scattering strength of a pressure-release sphere (dashed) and a rigid sphere (solid). . . 121 5.13 Magnitude of the scattering coefficients (in dB) for a spherical air

bubble in water: (a) uncloaked, and (b) cloaked using a single elas-tic layer with νc = 0.3 and

b

a = 1.30, designed to cancel the first 4 modes at kd,0a = 1.75, with the material properties listed in Ta-ble 5.4. The scattering gain (in dB) is given in (c), referenced to the scattering strength of a pressure-release sphere (dashed) and a rigid sphere (solid). . . 123 5.14 Real part of total pressure field for a spherical air bubble in water: (a)

uncloaked, and (b) cloaked using a single elastic layer with νc= 0.3 and b

a= 1.30, designed to cancel the first 4 modes at kd,0a = 1.75, with the material properties listed in Table 5.4. The color scale for the pressure is normalized to the amplitude of the incident wave, which is a time-harmonic plane wave traveling from bottom to top with a frequency of kd,0a = 1.75. . . 124 6.1 A time-harmonic incident plane wave in a fluid medium impinging on

an isotropic elastic core of radius a coated in two concentric shells of uniform thickness with outer radius b. The surrounding medium has density ρ0 and bulk modulus κ0, and the elastic core has density ρ, bulk modulus κ and shear modulus µ. . . 127

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6.2 Variation in cloaking layer density as a function of the shell thickness, for the (a) outer cloaking layer and (b) inner cloaking layer. The two layer fluid cloak is enclosing a rigid, immovable sphere at kd,0a = 1.0, with (a) δ2 = 0.01, (b) δ1 = 0.01. The cloaking layer densities are normalized by the density of the fluid in the surrounding medium, and the shell thickness is normalized by the radius of the inner sphere. Thin shell results are calculated using Equations (6.77) and (6.94). . 144 6.3 Variation in the bulk modulus of the outer cloaking layer κc1 as a

function of the (a) shell thickness δ1 and (b) inner cloaking layer bulk modulus κc2, enclosing a rigid, immovable sphere at kd,0a = 1.0. In (a) δ2 = 0.01 with κc2= 1, and (b) δ1 = δ2 = 0.01. The cloaking layer bulk modulus is normalized by the bulk modulus of the fluid in the surrounding medium, and the shell thickness is normalized by the radius of the inner sphere. Thin shell results are calculated using Equation (6.92). . . 146 6.4 Variation as a function of outer cloaking layer bulk modulus κc1 for

(a) the inner cloaking layer bulk modulus κc2 (b) the inner cloaking layer density ρc2 and (c) the outer cloaking layer density ρc1, enclos-ing a rigid, immovable sphere for δ1= δ2= 0.01 at kd,0a = 2.0. The cloaking layer properties are normalized by those of the fluid in the surrounding medium. Thin shell results are calculated using Equa-tions (6.81), (6.95) and (6.97). . . 148 6.5 Variation as a function of inner cloaking layer bulk modulus κc2for (a)

the outer cloaking layer bulk modulus κc1(b) the outer cloaking layer density ρc1 and (c) the inner cloaking layer density ρc2 for δ1= δ2= 0.04 at kd,0a = 2.0. Curves are shown for 4 different core materials: steel (solid red line), aluminum (solid blue line), glass (solid green line) and a rigid, immovable sphere (dashed black line). Material properties are listed in Table 4.1. The cloaking layer properties are normalized by those of the fluid in the surrounding medium, which is water. Thin shell results are calculated using Equations (6.77), (6.92) and (6.94). . . 150 6.6 Real part of the total pressure field for a steel sphere in water at

kd,0a = 2.0: (a) uncloaked, (b) cloaked using two fluid layers, and (c) cloaked using a fluid inner layer and an elastic outer layer. For the cloaked spheres, each layer of the cloak has a shell thickness of δ = 0.04. The color scale for the pressure is normalized by the amplitude for the incident wave, which is a time-harmonic plane wave impinging from bottom to top. . . 153 6.7 Scattering coefficients (in dB) for an (a) uncloaked and (b) cloaked

steel sphere in water. The cloak consists of two fluid layers with δ1= δ2= 0.04, which cancels the first three scattering modes at kd,0a = 2.0. The scattering gain in dB, relative to the uncloaked scatterer, is given in (c) for the exact theoretical solution and using finite elements (COMSOL). . . 155

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6.8 Variation as a function of outer cloaking layer Poisson’s ratio νc1 for (a) the outer cloaking layer bulk modulus κc1 (b) the inner cloaking layer density ρc1and (c) the outer cloaking layer density ρc2, enclosing a steel sphere in water for δ1= δ2= 0.04 at kd,0a = 2.0. The cloaking layer properties are normalized by those of the fluid in the surround-ing medium. Thin shell results are calculated ussurround-ing Equations (6.77), (6.92) and (6.94). . . 156 6.9 Scattering coefficients (in dB) for an (a) uncloaked and (b) cloaked

steel sphere in water. The cloak consists of an inner fluid layer and outer elastic layer with δ1= δ2= 0.04, which cancels the first three scattering modes at kd,0a = 2.0. The Poisson’s ratio of the outer elastic layer is 0.3. The scattering gain in dB, relative to the uncloaked scatterer, is given in (c) for the exact theoretical solution and using finite elements (COMSOL) for the case when the outer layer is an isotropic elastic solid. The scattering gain for the case of a fluid outer layer (dashed) is given for reference. . . 158 6.10 A time-harmonic incident plane wave in a fluid medium impinging on

an isotropic elastic core of radius a coated in four concentric shells of uniform thickness with outer radius b. The layers consist of alternat-ing materials C1 and C2, respectively, startalternat-ing with the outermost layer. The surrounding medium has density ρ0 and bulk modulus κ0, and the elastic core has density ρ, bulk modulus κ and shear modulus µ. . . 160 6.11 Scattering coefficients (in dB) for an (a) uncloaked and (b) cloaked

steel sphere in water. The cloak consists of four fluid layers, which cancels the first four scattering modes at kd,0a = 2.0. The cloaking layer properties are given in Table 6.2. The scattering gain in dB, relative to the uncloaked scatterer, is given in (c). . . 162 6.12 Scattering coefficients (in dB) for an (a) uncloaked and (b) cloaked

steel sphere in water. The cloak consists of a four layer cloak, consist-ing of alternatconsist-ing fluid and elastic layers, which cancels the first three scattering modes at kd,0a = 2.0. The Poisson’s ratio of the elastic layer is 0.3, and the cloaking layer properties are given in Table 6.2. The scattering gain in dB, relative to the uncloaked scatterer, is given in (c) for the case when the outer alternating layer is fluid (dashed) and elastic (solid). . . 163 6.13 Real part of the total pressure field for a steel sphere in water at

kd,0a = 2.0: (a) uncloaked, (b) cloaked using four fluid layers, and (c) cloaked using alternating fluid and elastic layers. The cloaking layer properties are given in Table 6.2. The color scale for the pressure is normalized by the amplitude for the incident wave, which is a time-harmonic plane wave impinging from bottom to top. . . 164

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7.1 Geometry of the anisotropic media considered in this chapter, for a material with (a) transverse isotropy, and (b) spherical isotropy. . . 167 7.2 Comparison of a two layer cloak with an inner fluid layer and (a) an

isotropic elastic outer layer, previously described in Chapter 6, and (b) an anisotropic (spherically isotropic) elastic layer. One possible way for creating the necessary anisotropy is illustrated in (b) using a compliant kerf filler, shown in black. . . 188 7.3 Scattering coefficients (in dB) for an (a) uncloaked and (b) cloaked

steel sphere in water. The cloak consists of an inner fluid layer and outer elastic layer with δ1= δ2= 0.04, which cancels the first three scattering modes at kd,0a = 2.0. The outer layer shown is spherically isotropic, with the same properties as the isotropic elastic case given in Table 6.1, except that the coefficient C11 in Equation 7.1 is scaled by a factor of 0.1. The scattering gain in dB, relative to the uncloaked scatterer, is given in (c) for the exact theoretical solution when the outer layer is a spherically isotropic elastic solid. The scattering gain for the case of a isotropic elastic outer layer (dashed) is given for reference. . . 189 C.1 (a) A time-harmonic incident plane wave in a fluid medium impinging

on a spherical core of radius a coated in a cloak of radius b, and (b) the equivalent configuration in COMSOL. . . 215

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Chapter 1 Introduction

For many years, the concept of cloaking has been a prevalent theme in sci-ence fiction and popular culture, from Start Trek to Harry Potter. The traditional approach taken by scientists and engineers to hide an object has been to apply an absorptive coating, thereby reducing the amount of energy reflected back to the source. Absorption alone, however, is limited by the physical properties of available lossy materials, and to compensate for this can require significantly increasing the thickness of the coating. Due to practical limitations in terms of size and weight, these absorptive layers are often not sufficient to eliminate the electromagnetic or acoustic ‘visibility’ due to the scattering from the object. Furthermore, even a per-fect absorber casts a shadow, which can provide a means of detection. It has not been until more recently that the topic of cloaking has received serious attention from the scientific community. This research has been driven by theoretical works describing how waves could be effectively bent around an object, or alternatively, by applying a coating to the object which effectively cancels the scattered field.

1.1 Thesis objectives

The objective of this thesis is to investigate the physical parameters nec-essary for, and the feasibility of, designing an acoustic cloak using a scattering cancellation approach for an elastic sphere using fluid and elastic layers. To achieve this objective, two fundamental questions will be addressed throughout this work:

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1. How can the acoustic field scattered from a spherical object be significantly reduced or cancelled?

2. How can a realizable cloak be achieved?

To address the first question, a thorough analysis of the existing body of scientific literature is required. For the case of acoustic cloaking, this includes pre-vious approaches to cloaking, including both electromagnetic (EM) waves and those involving acoustic waves in fluids and elastic solids. To gain a deeper understanding also requires a detailed look at the fundamental physics of scattering from spherical elastic bodies. With this foundation, the investigation of how the scattered acoustic field can be eliminated and the relevant physical parameters needed to accomplish this can be determined.

To address the second question, the relevant cloaking parameters will need to be considered. In particular, it is important to account for elastic effects, which are inherent to any practical acoustic structure. Although the elastodynamics of a submerged spherical body will lead to significantly increased complexity, thereby limiting the ability to obtain explicit analytic solutions, except for some limiting cases or specific examples. It is important to be mindful of how these particular cases fit into the broader context of acoustic cloaking, and therefore the solutions will be developed in the most generalized manner possible. Even when it is impractical or impossible to express the configurations explicitly, exact systems of equations can be solved numerically to determine the results in these instances. In fact, under some circumstances for which approximate solutions can be obtained, it will still be necessary to solve for the exact solution to precisely determine the effectiveness of a particular cloak design.

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1.2 Thesis overview

The chapters which follow have been laid out to address the objectives and research questions put forth in the previous section, and to convey the information in as clear a manner as possible. This process begins with a background on the basic theory and current approaches to cloaking, presented in Chapter 2. This includes approaches which have showed promise for EM or acoustic waves, examining how they work and identifying the potential limitations and challenges associated with each method.

To supplement this understanding of acoustic cloaking, a detailed formula-tion for the acoustic scattering from a submerged coated elastic sphere is developed in Chapter 3. Although classical scattering concepts have been studied extensively for over half a century, the vast majority of these works have focused on determining the scattered field for a given geometry and material properties, and the determina-tion of distinctive features based on these results which can enable identificadetermina-tion of the object [1]. Rather than using the scattered acoustic field for identification, the more recent concept of scattering cancellation developed for EM cloaking is applied, in which the scattered field in the surrounding medium is prescribed to be zero. Thus, instead of asking what coated objected caused this acoustic scattering?, the novel question considered here is what sort of coatings can produce zero or minimal acoustic scattering?.

The general formulations developed in Chapter 3 are analyzed in detail for particular configurations in Chapters 4, 5 and 6, presented in order of increasing complexity. In Chapter 4, the case of a submerged sphere coated with a single layer is investigated. Even in this simplest of cases, determining exact explicit expressions for the required cloaking layer properties is not feasible. Therefore, approximate solutions are developed for two different scenarios: (i) quasi-static limit, and (ii)

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thin shells relative to the wavelength of the incident sound. Based on these results, the effects of using a fluid or an isotropic elastic solid as a cloak is examined, for the cloaking of both fluid and elastic spheres. Chapter 5 presents a means to increase the effectiveness of the single layer cloak presented in Chapter 4. To do this, shear-wave anti-resonances within an isotropic elastic cloaking layer are used to increase the reduction in the scattering strength.

The presence of resonances in the single elastic cloaking layer presented in Chapter 5, however, can lead to a reduction in the bandwidth. Building upon the non-resonant single configuration presented in Chapter 4, Chapter 6 presents an investigation of the use of multiple layers. To facilitate this, the case of two layers is examined in detail. Although conceptually it is a relatively minor step to move from a single homogeneous layer to two homogeneous layers, the scattering analy-sis becomes significantly more cumbersome. Due to this, development of analytic expressions is limited to cloak consisting of two fluid layers. Although a relatively simple configuration, the results developed are used to investigate more complicated multilayer cloaks, by using two alternating materials.

The analysis in Chapter 6 allows for an arbitrary number of fluid or isotropic elastic solids to be used to construct a cloak. Although the operation of these cloaks is focused on a single design frequency, a reduction in the scattering strength is seen for a broader range of frequencies. To extend this bandwidth, consideration of anisotropic layers is considered, which is presented in Chapter 7. In spherical coordinates, the formulation developed in Chapter 3 and used in Chapters 4-6 cannot be used. An alternative approach to describe the scattering from an elastic sphere with anisotropic spherical shells is developed in Chapter 7, to which the conditions for scattering cancellation can be applied.

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remarks. Potential future applications of this work and expansions of the analysis developed are considered.

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Chapter 2 Background and fundamentals of acoustic cloaking

The topic of acoustic cloaking has gained significant attention in recent years, as the result of successful implementations in the electromagnetic (EM) domain. This chapter introduces several EM cloaking approaches which offer a means to achieving cloaking of acoustic waves. In Section 2.1, the transformation method is discussed, which prescribes anisotropic functionally-graded materials to re-route the incident energy around an object. This method has been the topic of much interest in the literature and has direct analogues in acoustics. In Section 2.2, another means of cloaking originally proposed for EM waves is described. This proposed method uses the anomalous resonances resulting from the interaction with negative-valued material properties to achieve cloaking. Finally, in Section 2.3, a promising method is discussed which utilizes a cloaking layer designed to eliminate the scattered field within the surrounding medium over a given bandwidth. This scattering cancellation approach has previously only been applied to the cloaking of EM waves. A qualitative comparison of the analogous behavior between EM and acoustic waves is presented, revealing that the main limitations and challenges encountered with implementation of this type of cloak for EM waves are mitigated in the case of acoustic waves.

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2.1 Transformation cloaks for acoustic waves

One proposed approach to cloaking is to utilize materials whose properties lead to the re-routing of sound around the object of interest. This approach prevents acoustic energy from interacting with the object, and was originally described for EM waves [2], [3]-[11] and subsequently extended to acoustic [12], [13]-[20] and elastic waves [21], [22]-[25]. To understand how this is possible, consider an undisturbed acoustic wave traveling through a homogeneous medium. To achieve cloaking, the objective is to create an annulus-shaped region that is deformed in such a way as to preserve the undisturbed acoustic wave outside of this enclosed region. Thus, while outside of this deformed region the acoustic wave is undisturbed, there is no acoustic field present within the interior of the annulus, and therefore allows any object in this region to be cloaked.

Mathematically, this is accomplished by using a one-to-one mapping between the deformed and undeformed regions everywhere except at a single point, which is mapped onto the cloak’s inner boundary [26]. This process is illustrated in Fig-ure 2.1. Note that this process, which is formally called a coordinate transformation, alters the cloak’s properties so that the modified wave equation in the cloak mim-ics the wave equation in the previously unaltered region. Due to the mathematmim-ics defining cloaks employing this approach, it is referred to here as the transformation method.

Mathematically, the general representation for relating the wave equation in the initial and transformed spaces is given by Norris [21]

∇2p − ¨p = 0 ⇐⇒ KS : ∇ ρ−1S∇p − ¨p = 0. (2.1) In the undeformed space, this relationship is simply the wave equation for a ho-mogeneous fluid. The expression for the deformed space is more complicated, and

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b

a b

Figure 2.1: Illustration of the coordinate transformation between an undeformed space (left) and a symmetric deformed space (right). Any object placed within the interior region of radius a will be cloaked.

involves spatial variability of both the pressure p and a generalized material prop-erty called the density tensor ρ. The parameters K and S describe the stress-strain relationship in the transformed region, which is σ = Cε, where

C = KS ⊗ S. (2.2)

In this relationship, the second order tensor S is the result of the transformation and satisfies the condition div S = 0. Note that the : and ⊗ in the above expressions denote the inner and outer tensor products, respectively. Two different types of cloaks arise from this transformation: inertial cloaks and pentamode material cloaks, which are explored in the next two sections.

2.1.1 Inertial cloak

The analysis of determining the necessary coordinate transformation with general elastic media and arbitrary geometries can be formidable. One case of interest, which was the first form of acoustic cloaking to be explored by Cummer

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et al. [12], is when the compressibility is isotropic, like that of a fluid. This is equivalent to the case where S is equal to the identity matrix in Equations (2.1) and (2.2). The basic formulation for this case is presented in Section 2.1.1.1. Based on these theoretical values for the cloaking layer properties, Section 2.1.1.2 discusses current approaches to the realization of these cloaks.

2.1.1.1 Basic formulation

Simplification of the solution to the transformation in Equation (2.1) occurs when an isotropic compressibility is considered for canonical shapes in a rotation-ally symmetric coordinate system, like that shown in Figure 2.2(a). With these simplifications, expressions for the transformed cloaking layer properties become [21] ρr ρ0 = R0 r R N −1 , (2.3) ρθ ρ0 = 1 R0 r R N −3 , (2.4) κ κ0 = 1 R0 r R N −1 , (2.5)

where ρr and ρθ are the radial and azimuthal densities of the cloak, κ is the bulk modulus of the cloak, ρ0and κ0are the density and bulk modulus of the surrounding fluid, and N is the number of spatial dimensions. For solutions of practical signifi-cance, N = 2 or N = 3, corresponding to either cylindrical or spherical coordinates, respectively. The function R appearing in Equations (2.3)–(2.5) is the radius in the undeformed space shown on the left-hand side of Figure 2.1, which is related to the radius r in the deformed space by

R = b

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(a) (b)

−1 0 1

Figure 2.2: (a) An incident plane wave impinging upon a cylindrical body covered with an inertial cloak with density tensor ρ and bulk modulus κ. (b) Numerical simulation from Cummer et al. of the total pressure field for a perfect inertial cloak [14]. The amplitude is normalized by the magnitude of the incident wave, which is a time-harmonic plane wave traveling from left to right.

The function R0 denotes the first derivative of R with respect to r, which is simply R0 = b

b − a. (2.7)

Notice that the values of r = a and r = b in the right-hand side of Figure 2.1, representing the inner and outer edge of the cloak, correspond to the values R = 0 and R = b in the undeformed space shown on the left-hand side of Figure 2.1.

Substituting Equations (2.6) and (2.7) into Equations (2.3)–(2.5), one can obtain the required physical properties necessary to achieve acoustic cloaking for a spherical (or cylindrical) object of radius a with an acoustic cloak of outer radius b

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[14] ρr ρ0 = b − a b N −2 r r − a N −1 , (2.8) ρθ ρ0 = b − a b N −2 r r − a N −3 , (2.9) κ κ0 = b − a b N r r − a N −1 . (2.10)

Since the wave speed in a fluid is given by c = qκ_ρ, the radial and azimuthal wave speeds can be written as

cr c0 = b − a b , (2.11) cθ c0 = b − a b r r − a , (2.12)

where c0 is the wave speed in the surrounding fluid. One of the defining character-istics of this type of cloak is the requirement that the density is anisotropic. Due to this unusual inertial effect, cloaks of this type are referred to as inertial cloaks [21]. Examining Equations (2.8)–(2.10), one finds these cloak properties possess a similar form, consisting of a scaling term proportional to (b − a)/a and a radial dependence proportional to r/(r−a). Although the exact functional dependence of ρr(r), ρθ(r) and κ(r) is different for these two coordinate systems, it is noted from Equations (2.11) and (2.12) that the radial and tangential wave speeds within the cloak are independent of the number of spatial dimensions. In particular, the radial wave speed is independent of r, and simply scales in magnitude depending on the values of the inner and outer radii, a and b. Since the wavelength is proportional to the wave speed for a given frequency, this means that the wavelength scales such that there are the same number of wavelengths over a distance a in the surrounding fluid as there are in the distance (b − a) in the cloak. Furthermore, Equations (2.8)

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and (2.11) reveal that the radial specific acoustic impedance, ρrcr, is equal to that of the surrounding medium at the outer edge of the cloak.

A numerical example of the resulting pressure field obtained for an inertial cloak is given in Figure 2.2(b). In this figure, the compression of the radial wave-fronts prescribed by Equation (2.11) is clearly observed. At the inner edge of the cloak near r = a, it is apparent that the phase front is the same on all sides of the object simultaneously. For this to occur, the tangential phase speed must become infinitely large. This is clear from inspection of Equation (2.11) which requires that cθ → ∞ as r → a. In the limit of r = a, this would present a clear violation of causality. Another significant problem that arises in both cylindrical and spherical coordinates is that the density and the bulk modulus of the cloak must become infinite at the inner edge of the cloak.

Despite the apparent physical requirements of anisotropic inertia, non-causal wave speeds and infinite mass, extensive research has been performed with the aim of developing a practical inertial cloak. To address these limitations, the primary focus has been on cloaks which allow a small but finite amount of scattering, which are sometimes referred to as near perfect cloaks [21] (though often times this distinction is not addressed in the literature).

For a perfect cloak, recall that the coordinate transformation creates a re-gion which does not interact with the incident wave, so that any object can be placed inside without affecting the cloaking condition. Using a near perfect cloak, however, will inevitably allow some energy to penetrate into the interior. In this case, the effectiveness has been observed to depend on the properties of the object being cloaked, particularly near resonance frequencies of the object [13, 27]. An-other challenge with inertial cloaking is dealing with the radially inhomogeneous (functionally-graded) properties of the cloak, which vary as a continuous function

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of r. Overcoming this difficulty is accomplished by dividing the cloak into discrete, homogeneous layers [27].

For an inertial cloak, there is the unique problem of producing a material with anisotropic inertia, a property which can be achieved by utilizing the effective properties of a periodically layered fluid system. The characterization of this behav-ior, and the development of explicit expressions relating the different components of the effective density to the constituent fluid layers, is credited to Schoenberg and Sen [28]. Interestingly, this work was published well before there was serious scientific discussion of cloaking, and was presented with regards to the acoustical interactions of the ocean bottom. Originally derived for periodic fluid layers in a half-space, these results have been applied directly by assuming that the curved discrete layers within the cloak are sufficiently thin. Assuming two alternating fluids denoted by the subscripts 1 and 2, the resulting homogenized properties are

ρr= 1 d1+ d2 d1ρ1+ d2ρ2, (2.13) 1 ρθ = 1 d1+ d2 d1 ρ1 +d2 ρ2 , (2.14) 1 κ = 1 d1+ d2 d1 κ1 + d2 κ2 , (2.15)

where dm is the thickness of the mth layer.

To address the issue of creating anisotropic inertia, Equations (2.13)–(2.15) provide a useful way for obtaining discrete layer properties using fluid layers. Un-fortunately, the constitutive fluids needed to match the properties given in Equa-tions (2.8)–(2.10) must also exhibit similarly extreme properties, and most proposed layering schemes require a mixture of extremely dense and extremely light fluids rel-ative to the surrounding medium. As a result, inertial cloaks made from alternating fluid layers are impractical to construct and, for the most common ambient fluids

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encountered (namely air and water), fluids possessing these vast material property contrasts do not exist in nature. Configurations using three fluid sublayers have been proposed by Norris and Nagy [29], which results in sublayers with low den-sities, near-neutral denden-sities, and high densities. These three layer cases, however, also require the use of fluids which currently do not exist in order to achieve cloaking.

2.1.1.2 Use of acoustic metamaterials

Although homogeneous fluids are impractical to use in the construction of inertial cloaks, recent research has focused on creating fluids which exhibit the prop-erties required. In particular, attention has been focused on acoustic metamaterials, which are materials containing organized microstructures that create extreme effec-tive macroscopic properties that are difficult or impossible to achieve using ordinary materials. Acoustic metamaterials can be thought of as a type of composite struc-ture, although they have the distinct feature of exhibiting effective properties beyond the normal bounds of an ordinary homogenized mixture.

Despite their exotic nature, acoustic metamaterials have been realized using basic physical mechanisms, including the lumped-element behavior of a series of ports and cavities by Zhang et al. [30] and the multiple scattering effects of a lattice of cylindrical scatterers by Torrent and S´anchez-Dehesa [16]. More recent efforts by Popa et al. [31] have used more complex engineered structures, but these fundamentally rely on the same basic principles of lumped-element and scattering effects within each structural unit, in addition to the periodic arrangement of these units, to achieve the desired extreme macroscopic properties.

Figure 2.3(a) shows one type of acoustic metamaterial for 2D acoustic cloak-ing applications proposed by Torrent and S´anchez-Dehesa [15]. In this configuration, each layer is made up of two fluid sublayers (shown in red and blue in the figure),

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(a) 1000 100 10 1 0.1

1E−4 1E−2 1 1E2 1E4

ρ_eff c eff

(b)

Figure 2.3: (a) Schematic view of a cloaked cylinder proposed by Torrent and S´anchez-Dehesa [15]. The cloak consists of alternating fluid sublayers of equal thick-ness. (b) Parametric plot showing the range of material properties needed to achieve cloaking using a configuration like that shown in part (a). The solid lines show the two fluid sublayer properties, and the dashed lines show the design space using a sonic crystal arrangement.

as described by Equations (2.13)–(2.15). To obtain the necessary fluid properties, each sublayer consists of a lattice of elastic scatterers. Utilizing the homogenized properties from this lattice (including multiple scattering effects) creates an acous-tic metamaterial commonly referred to as a sonic crystal [16]. Through the use of sonic crystals, a more realistic design space can considered, as illustrated by the parametric plot presented in Figure 2.3(b). In this figure, the solid lines represent the necessary values for the fluid sublayers based on Equations (2.13)–(2.15), while the dotted lines represent the parameter space which can be used to achieve these values with a sonic crystal.

This noticeable increase in the range of material properties which can be used arises from the variation in the composition and volume fraction of elastic scatterers within each fluid sublayer. Although the choice of using elastic cylinders with a fluid layer is partially responsible for such a broad expansion of the parameter space shown in Figure 2.3(b), the range of achievable densities are extended beyond the

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(a) (b) −1 0 1

Figure 2.4: Numerical simulations of the total pressure field for a cloaked rigid cylinder, using an inertial cloak consisting of alternating fluid sublayers containing sonic crystals by Torrent and S´anchez-Dehesa [15], with (a) 50 layers, and (b) 200 layers.

traditional bounds of composite mixtures through the judicious design of the lattice arrangement of the elastic scatterers. Such an effect was verified experimentally by Torrent et al. [32], achieving the effective properties of argon gas using a closely packed lattice of wooden cylinders in air.

Despite the promising characteristics that sonic crystals offer, there are still significant obstacles with regards to implementation in a practical inertial cloak. Figure 2.4(a) and (b) show numerical simulations of the total acoustic pressure field using sonic crystals for 50 layers and 200 layers, respectively. Even with discretiza-tion of the cloak into 50 layers, there is still significant visible disrupdiscretiza-tion of the field, even in the idealized case illustrated which does not include any losses.

Aside from the complexity of the structure, there are several other factors that significantly affect the realization of inertial cloaks. From Figure 2.3(b) it is observed that there are two distinct properties exhibited by the two sublayers, one

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(a)

(b)

(c)

Figure 2.5: Experimental design by Zhang, Xia and Fang [30] for a cloaked steel cylinder in water, using an inertial cloak with 16 discrete layers of a radially sym-metric lattice of ports and cavities, for (a) the entire cloak, and (b) a close up to show the arrangement of ports and cavities. The measured pressure field passing through the shadow zone of the rigid cylinder are shown in (c) for the cloaked (blue) and uncloaked (red) configurations, relative to a freefield measurement (green).

which is consistently lower in density than the surrounding fluid, and the other which is consistently heavier. Since the required cloak properties are normalized based on the surrounding fluid properties, the relative difficulty in realizing the necessary cloak depends on the particular properties of the surrounding fluid, and can be addressed by varying the filling fraction of each sublayer [33]. However, the most commonly encountered surrounding media in acoustic applications are air and water, which lie on opposite extremes of the range of material densities, potentially limiting the effectiveness of what cloaking layer properties can be achieved, even with an arbitrarily large number of layers.

Another type of acoustic metamaterial which has been developed utilizes transmission-line arrangements of acoustic lumped elements made up of ports and

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cavities. Arrangements of acoustic lumped elements have been used for a cylindrical configuration, made up of discs that were stacked together, an example of which is shown in Figure 2.5(a) based on the work of Zhang, Xia and Fang [30]. A close up is shown in Figure 2.5(b), in which the individual ports and cavities can be clearly observed.

To design this cloak, the features illustrated in Figure 2.5(a) and (b) were machined out of aluminum discs, using the surrounding fluid (in this case water) to fill the resulting ports and cavities. To achieve anisotropic inertial effects, different sized ports were used in the radial and tangential directions while using a shared cavity. The cavity sizes were the same for each layer, but were allowed to change size with the radial distance from the inner cylinder.

Using the prescribed properties for ρr, ρθ and κ for an inertial cloak, expres-sions for the acoustical inductance in the radial and tangential directions, Lr and Lθ, and the acoustical compliance C become [30]

Lr= ρw lr Sr , (2.16) Lθ = ρw lθ Sθ r − a r 2 , (2.17) C = V Kw a b − a 2 , (2.18)

where l is the port length, S is the cross-sectional area of the port, V is the volume of the cavity, and ρwand κw are the density and bulk modulus of water, respectively. Note that the sound speed for a transmission-line configuration of acoustic lumped elements is [34]

c = r

1

LC. (2.19)

Using Equations (2.16)–(2.18), the sound speed given by Equation (2.19) can be determined for the radial and tangential directions and equated with the prescribed

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values given by Equations (2.11)–(2.12).

Figure 2.5(c) highlights typical data obtained from experimental results us-ing this design. In this plot, the total pressure field was measure in the shadow zone directly behind the target. Note that the axis of symmetry along which the center of the target lies is located at x = 150 mm based on the data illustrated. To interpret the results presented in Figure 2.5(c), a scale in decibels has been added to right-hand side which gives a measure of the sound pressure level (SPL) relative to the freefield levels. Comparing the SPL, it can be seen that the change due to the addition of the cloaking layer is about 3 dB or less over most of the range, with a maximum change of about 12 dB. Even when the cloaking layer is present, there is still a difference about 6 dB in the shadow zone compared with the freefield measurements. Considering that the thickness of the cloak is three times the radius of the target and contains approximately 3000 helmholtz resonators, it is uncertain how much of these observed changes in the data with the cloak present are due to thermoviscous losses.

There are still more significant limitations of designs using this type of acous-tic metamaterial. First, since water (or any ambient fluid) is used as the medium through which sound propagates, this limits how fast the information can propagate within the cloak. Although the high phase speeds needed can be achieved once the acoustic field reaches steady state, this means that the performance with transients will be significantly diminished. Furthermore, one design aspect which appears to have not been considered by Zhang et al. is the end correction of the ports to ac-count for the mass of the entrained fluid [34]. Based on the design presented, thin wide ports are used to achieve the low inertia needed for the high phase speeds near the inner edge of the cloak. However, when the end corrections are properly accounted for, this significantly reduces the achievable phase speeds.

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2.1.2 Pentamode materials and acoustic metafluids

To address the requirement for infinite mass necessary at the inner edge to achieve perfect cloaking with an inertial cloak, alternative solutions have been sought based on the general coordinate-transformation given by Equation (2.1). By assuming that the density tensor can vary with radial position, but is isotropic at any point in space, the transformed cloak properties are [21]

ρ ρ0 = R0 R r N −1 , (2.20) κr κ0 = 1 R0 R r N −1 , (2.21) κθ κ0 = R0 R r N −3 , (2.22)

where R and R0 represent the radius in the undeformed space and its derivative, given by Equations (2.6) and (2.7), respectively. By prescribing isotropic density in this case, the anisotropy required to re-route incoming waves around the cloaked space is achieved through the bulk modulus, with radial and tangential components κr and κθ, respectively.

Under these conditions, the stress-strain relations simplify to a single ex-pression. Note that the tensor C containing the elastic coefficients is 4th order, though due to symmetry contains at most 21 independent coefficients for a general anisotropic elastic body [35]. As a result, the stress-strain relation is often written in a more compact form in which C is expressed as a 6 × 6 symmetric matrix where each column of a row relates a component of stress to one of six possible defor-mations, given by the eigenmodes. Therefore, of the six different possible ways of deforming an elastic body, the type of material needed for this type of cloak does not resist deformation in five of these modes. Another way of saying this is that no

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(a) (b)

Figure 2.6: (a) Example of an acoustic metafluid by Norris [21], consisting of lubri-cated elliptical beads arranged in a hexagonal lattice. (b) A pentamode cloak using a proposed acoustic metafluid called metal water [37].

energy is required to deform the material in these modes. As a result, this type of material is known as a pentamode material [36].

The simplest example of a pentamode material is a fluid which does not resist deformation in five shear modes and only resists pure compression [21]. How-ever, conventional fluids have isotropic compressibility and therefore cannot satisfy Equations (2.20)–(2.22). Pentamode materials which do satisfy the conditions for the transformed cloak are called acoustic metafluids [38].

Figure 2.6 shows two acoustic metafluids proposed by Norris. In Figure 2.6(a), the acoustic metafluid is created using a microstructure of lubricated elliptical beads in a hexagonal lattice [38]. Use of non-spherical beads ensures that the macroscopic properties will exhibit anisotropy, as prescribed by Equations (2.20)–(2.22). An-other proposed acoustic metafluid is illustrated in Figure 2.6(b), for use as a cloak in water. The material for this cloak is made up of an air-filled, hinged metal lat-tice, so that an undeformed block will have the same density and bulk modulus as water, which has appropriately been called metal water [37]. A cloak can be created by physically deforming a cylinder (or sphere) of metal water to achieve a hole at the center with the desired radius. The deformation of the metal water represents the physical embodiment of the mathematical steps involved with the coordinate

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transformation. This will lead to denser portions exhibiting strong anisotropy at the inner part of the cloak as prescribed by Equations (2.20)–(2.22), and observed in Figure 2.6(b).

Although the use of acoustic metafluids addresses many of the issues with inertial cloaks, challenges remain with the construction and implementation of such materials. Effects of small amounts of resistance in some of these modes, such as that which could occur in real hinges, has yet to be determined. Furthermore, due to the condition of having five modes which do not resist deformation, designing such a structure while ensuring sufficient stability is a significant issue.

2.2 Anomalous resonance cloaking

In addition to the transformation-based approach discussed in the previous section, cloaking effects can occur through other physical means. One such type is cloaking by anomalous resonances, which has been theorized based on the analysis of a device known as a superlens, which was devised to permit sub-wavelength electro-magnetic imaging [39]. Superlens work by using materials with negative properties (either permittivity or permeability for electromagnetic waves), causing scattered evanescent waves to emanate from an illuminated surface and can therefore enable resolution beyond the tradition diffraction limits. This apparent evanescent wave ’amplification’ is referred to as an anomalous resonance.

Based on the theoretical formulations for a cylindrical electromagnetic su-perlens, it was observed by Nicorovici et al. that transparency could also occur, analogous to planar mirroring effects from a material with negative properties [40]. Due to this mirroring effect, cloaking with anomalous resonances occurs in the region external to the cloak itself, unlike the transformation-based approach that occurs within the interior of the cloak [39]. Figure 2.7(a) shows a numerical simulation

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(a)

(b)

(c)

Figure 2.7: (a) Numerical simulation by Nicorovici et al. [39] of the electric potential field demonstrating anomalous resonance cloaking. The inner and outer edge of the cloak are denoted by solid lines, and the region of cloaking is denoted with a dashed line. (b) Layout of a 1D mass-spring system by Fang et al. [41] experimentally shown to exhibit negative effective mass. (c) Proposed resonant inclusion for an acoustic anomalous resonance cloak by Zhou et al. [42]. In this illustration, red denotes upward displacement and blue denotes downward displacement.

from Nicorovici et al. [39] of the electric potential field for an electromagnetic cloak using anomalous resonances. In this figure, resonances can clearly be seen near the inner and outer edge of the cloak, though the cloaking effect occurs beyond the edge of the cloak and the associated anomalous resonances.

One of the difficulties associated with this approach is that it requires nega-tive material properties. For the acoustic analog to anomalous resonance cloaking, research has currently been focused on overcoming this particular challenge. Al-though negative mass or compressibility are not readily found in nature, the use of resonant elements within an acoustic material has been demonstrated to produce an effective negative mass.

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transmission-line arrangement of Helmholtz resonators operating near resonance [41]. This was experimentally demonstrated using a linear arrangement of the resonators along the length of a waveguide, shown in Figure 2.7(b), creating a 1D mass-spring system which exhibited effective negative mass for a small range of excitation frequencies. This is a similar approach taken to the one proposed for the inertial cloak using a lattice of ports and cavities shown in Figure 2.5(a) and (b). It is important to note that for the inertial cloak, the required effective mass is positive, whereas for the anomalous resonance it is negative, which can only be achieved over a very narrow frequency band near resonance. Another proposed method by Zhou et al. utilizes an acoustic metamaterial consisting of coated spherical inclusions [42]. The coated sphere is tuned to resonate so that the outer layer is moving out-of-phase with the sphere, as illustrated in Figure 2.7(c), thereby creating a net negative effective mass. To achieve the necessary negative material properties for anomalous reso-nance cloaking of acoustic waves, acoustic metamaterials with resonant structures are required. However, the use of such resonances are highly sensitive to real world factors, such as the internal losses and perturbations from the design specifica-tions encountered during the fabrication process [43]. Furthermore, to achieve these negative effective properties requires highly resonant systems which are inherently narrowband, therefore presenting a significant challenge in achieving a realizable broadband cloak.

2.3 EM plasmonic cloaking

An alternative approach which has been investigated extensively for electro-magnetic waves is the use of a coating which cancels the scattered field from the object, thereby achieving cloaking. This corresponds to mathematically solving the classical scattering problem, often referred to as Mie scattering for electromagnetic

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waves, for the coated object, and then solving to determine what coating properties will produce a null in the scattered field under a given set of conditions. As a result, this is typically referred to as a scattering cancellation approach.

Since it is only the scattered field at the outer edge of the cloak which is eliminated, there is no restriction on the interaction of the incident wave with the object being cloaked, as observed in Figure 2.8(a). This is conceptually different from the transformation method presented in Section 2.1, for which the incident wave is re-routed around the object to prevent any interaction with it. Previous work by Al`u and Engheta [43] has shown that scattering cancellation can occur using non-resonant or non-resonant wave interactions within the cloaking layer, with either positive or negative cloak properties to achieve this effect. This differs from the anomalous resonance method described in Section 2.2, which used the anomalous resonances resulting from the negative cloak properties combined with the positive properties of the surrounding medium to achieve cloaking at a distance away from the edge of the cloak. With scattering cancellation, the cloaking effect occurs everywhere in the surrounding medium, even in the case where negative properties are used.

To enable the reader to examine the scattering cancellation approach in more detail, the basic theory which has been developed for electromagnetic waves is summarized in Section 2.3.1. In addition, the type of material properties which are necessary will be discussed in Section 2.3.2. Based on this analysis of the EM scattering cancellation, an overview of how this relates to the possibility of being applied for acoustics waves is presented in Section 2.3.3.

2.3.1 Basic formulation

For electromagnetic waves, the general scattering problem may be solved using the Mie expansion technique. The fields scattered by an arbitrary object

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(a) (b)

Figure 2.8: (a) Demonstration of EM scattering cancellation using a plasmonic cloak [44], originally developed by Al`u and Engheta [45]. (b) Parametric plot for the cloaking layer properties of a single layer cloak utilizing EM scattering cancellation for a magnetodielectric sphere [44]. The color scale represents the scattering strength relative to the uncloaked scatterer.

subject to a time-harmonic excitation, which in this work are described using a e−iωt convention, may be represented by a sum of electric (ES) and magnetic (HS) outgoing waves, often referred to as transverse electric (TE) and transverse magnetic (TM), expanded in terms of spherical harmonics as [46]

ES= ∞ X n=1 n X m=−n cTM_nmbnm∇×∇×(rψmn) + iωµ0 ∞ X n=1 n X m=−n cTE_nmdnm∇×(rψmn) , HS= ∞ X n=1 n X m=−n cTE_nmdnm∇×∇×(rψmn) − iωε0 ∞ X n=1 n X m=−n cTM_nmbnm∇×(rψnm) , (2.23)

where ψ_nm is a spherical potential function, ε0 and µ0 are permittivity and perme-ability of the background, k0 = ω

√

ε0µ0 is the background wavenumber, b and d represent the known amplitudes of the spherical expansion of the impinging wave, and the scattering coefficients c depend on the geometry of the scatterer and on the frequency of operation. For the case of a spherical scatterer enclosed in a spherical

Acoustic cloaking of spherical objects using thin elastic coatings

Table of Contents

List of Tables

List of Figures

Chapter 1

Introduction

Chapter 2

Background and fundamentals of acoustic cloaking